The second way of generating acceleration of the expansion of the Universe is by modifying the laws of gravity at cosmological scales. Such a modification is not only motivated by accelerated expansion, but also by observations at galactic scales. The amount of visible matter (stars, gas, ...) that we detect directly seems indeed not sufficient to explain gravitational interactions in structures like galaxies and clusters of galaxies [33]. As for the problem of acceleration of the Universe, disagreement between observations and theoretical calculations can be explained either by invoking the presence of a new form of matter, called dark matter, or by modifying the laws of gravity at galactic scales. It is then appealing to look for a modified theory of gravity which could explain discrepancies at both galactic and cosmological scales. Modifications of gravity can be achieved in a covariant way either by modifying the form of the Einstein-Hilbert action, like in f(R) theories [34], or by introducing extra fields in the action which alter (directly or indirectly) the relation between the metric and the matter content. An example of modified theory of gravity containing an scalar field is Brans-Dicke theory [35]. Einstein-Aether theories [36] require the presence of a vector field, whereas TeVeS Bekenstein theory [37] contains a scalar and a vector field in addition to the metric field. Hence a plethora of modified theories of gravity have been constructed, which all have to satisfy observational constraints. In our paper [5] which is reproduced in Chapter 6, we study the compatibility of one class of theories, the generalized Einstein-Aether theories, with solar system experiments.
Generalized Einstein-Aether theories are characterized by the presence in the action of a timelike vector field coupled to gravity. The primordial motivation of this vector field was to introduce spontaneous breaking of Lorentz invariance, leading to variations in the speed of light. It has then been shown that such a dynamical vector field could generate modifications of gravity. Contrary to Einstein-Aether theory, in which the vector field possesses only quadratic kinetic terms, the generalized version allow for non-canonical kinetic terms. The interest of those lies in the possibility to modify the laws of gravity at both galactic and cosmological scales. Generalized Einstein-Aether theories provide consequently a promising alternative theory of gravity. As with General Relativity, they must therefore satisfy the constraints one infers from precise observations in the solar system. In our work we study solutions of generalized Einstein-Aether theories in the solar system. We use then solar system tests to place constraints on parameters of the theories.
At present, mainly three accurate observations allow to test the laws of gravity in the solar system [38]. The first one concerns the perihelion shift of Mercury. The orbit of Mercury is indeed not exactly an ellipse, since it does not close completely. Consequently, the perihelion of the orbit rotates slowly around the Sun. Newton’s law of gravity does not explain this shift, which is however predicted by General Relativity. The extremely good agreement between the observed and the predicted shift value has been a very strong evidence in favor of General Relativity. The second observation is related to deflection of light. One can measure, during solar eclipses, deflection of rays emitted by stars located behind the Sun. Calculations in Newtonian’s gravity predict only half of the observed deflection, whereas General Relativity produces a completely compatible value. Finally, the third observation uses the time delay of radio signals passing through a gravitational
potential to test gravity in the solar system. One sends a signal from the earth toward a planet or a satellite situated at the other side of the Sun, and then observes the reflected signal. General Relativity predicts a time delay of the signal with respect to the Newtonian time, due to its trajectory through the gravitational potential of the Sun. The observed time delay is in excellent agreement with predictions of General Relativity. Hence when modifying General Relativity and constructing a new theory of gravity, one needs to satisfy these three observational constraints provided by the solar system. Observations require therefore a theory which reproduces General Relativity at solar system scales but differs substantially at galactic and cosmological scales.
In our work, we investigate the behavior of generalized Einstein-Aether theories in the Solar System. We study spherically symmetric and static solutions of the equations of motion. We work, as a first step, within the framework of the Post-Newtonian Parame-terization (PPN). This method proposed by Nordtvedt [39] and generalized by Will [38]
uses spherically symmetric and static solutions of General Relativity as a guide for solu-tions in modified theories of gravity. Einstein’s equasolu-tions can indeed be solved exactly in the solar system, providing the so-called Schwarzschild solution. This is a decreasing function of the distance from the Sun r. Hence one can calculate its weak field limit, by expanding it in inverse powers of r. Solar system tests provide constraints on the first coefficients of this expansion. The idea of PPN is then to replace numerical coefficients of Schwarzschild expansion by parameters and to use it as an ansatz solution for modified theories of gravity. One can then solve the modified equations of motion which determine the parameters. Those are called the Post-Newtonian parameters. A modified theory of gravity is then compatible with solar system observations only if its Post-Newtonian pa-rameters are in agreement with the values inferred from observations. However, in the case of generalized Einstein-Aether theories, we show that the Post-Newtonian ansatz is not a physically acceptable solution of the equations of motion. This ansatz forces indeed one parameter (c1) of the generalized Einstein-Aether Lagrangian to vanish. However it has been shown that c1 must be strictly negative in order for the theory to possess a positive definite Hamiltonian [40]. We therefore conclude that spherically symmetric and static solutions of generalized Einstein-Aether equations can not be expanded in negative powers of distance from the Sun, except in the special case where the non-canonical kinetic terms reduce to a quadratic one, i.e. in usual Einstein-Aether theory.
We then study a generalization of the Post-Newtonian expansion by adding positive powers of the distance from the Sun. Mathematically, such an expansion makes sense since positive and negative powers define a complete basis. Physically, positive powers are motivated by the dark matter problem. They are indeed required in order to generate an increase of the gravitational potential at large distances from the gravitational source.
However, those terms are generally neglected in the solar system. Coefficients in front of positive powers of r must indeed be very small in order not to interfere with solar system tests. One consequently assumes that they do not play any role in the equations of motion at solar system scale [41, 42, 43]. This is however not correct for generalized Einstein-Aether theories, where positive powers modify the equations in a crucial way, even if their associated coefficients are small. This follows from the presence of a mass parameter in the Lagrangian, whose value is fixed in order to modify gravity at galactic scales. This mass parameter is then small enough to mix the large coefficients in front of
the negative powers with the small coefficients in front of the positive powers, and as a result to modify the equations of motion in a measurable way. In other words, in order to modify gravity at galactic scales generalized Einstein-Aether theories require the presence of a small mass parameter in the Lagrangian. This affects the equations of motion at solar system scales in such a way that positive powers of r are no more negligible. An interesting consequence of positive powers in the expansion is the fact that the parameter c1 of the Lagrangian does not vanish any more and consequently that the Hamiltonian can be positive definite. Furthermore, using positive and negative powers of r, we find a set of solutions compatible with solar system observations. Hence generalized Einstein-Aether theories are in agreement with solar system constraints and are consequently viable alternative theories to General Relativity.
Another important aspect of our work on generalized Einstein-Aether theories is the method we developed in order to prove that an ansatz solution satisfies the equations of motion at each order. It is indeed not sufficient to calculate the Post-Newtonian parameters and to compare them with measured values to show that a solution is compatible with solar system constraints. We need also to be sure that the following coefficients in the expansion do not lead to any inconsistency. In our work we present a method to test whether an expanding solution satisfies the equations of motion at each order. The interest of this method is that it requires only to calculate some of the first coefficients of the expansion, as well as their corresponding equations, to prove that the complete expansion satisfies the equations of motion. It is consequently not needed to solve each order explicitly. Our method is valid for expansion in positive and negative powers of r and is therefore very general.
Fluctuations of the luminosity
distance
PHYSICAL REVIEW D73, 023523 (2006)
Fluctuations of the luminosity distance
Camille Bonvin, Ruth Durrer and M. Alice Gasparini
We derive an expression for the luminosity distance in a perturbed Friedmann universe. We define the correlation function and the power spectrum of the luminosity distance fluctuations and express them in terms of the initial spectrum of the Bardeen potential. We present semi-analytical results for the case of a pure CDM (cold dark matter) universe. We argue that the luminosity distance power spectrum represents a new observational tool which can be used to determine cosmological parameters. In addition, our results shed some light into the debate whether second order small scale fluctuations can mimic an accelerating universe.
DOI: 10.1103/PhysRevD.73.023523 PACS numbers: 98.80.-k, 98.62.En, 98.80.Es, 98.62.Py
2.1 Introduction
Some years ago, to the biggest surprise for the physics community, measurements of lumi-nosity distances to far away type Ia supernovae have indicated that the Universe presently undergoes a phase of accelerated expansion [6, 7, 44]. If the Universe is homogeneous and isotropic, i.e. a Friedmann-Lemaˆıtre universe, this means that the energy density is domi-nated by some exotic ’dark energy’ which obeys an equation of state of the formP <−ρ/3.
The best known dark energy candidate is vacuum energy or, equivalently, a cosmological constant. This discovery has lately been supported by several other combined data sets, like the cosmic microwave background (CMB) anisotropies combined with either large scale structure or measurements of the Hubble parameter [45].
On the other hand, since quite some time, it is known that locally measured cosmological parameters likeH0or the deceleration parameterq0might not be the ones of the underlying Friedmann universe, but they might be dressed by local fluctuations [46, 47, 48]. Therefore, it is of great importance to derive a general formula of the luminosity distance in a universe with perturbations. To some extent, this has been done in several papers before [49]-[57]
But the formula which we derive here is new. We shall comment on the relations later on.
Lately, it has even been argued that second order perturbations might be responsible for the observed acceleration and that no cosmological constant or dark energy is needed [58]-[62]. This claim is very surprising, as it seems to require that back reaction leads to big perturbations out to very large scales, contrary to what is observed in the CMB. This proposal has thus promptly initiated a heated debate [63]-[66].
On the one hand, the present work is a contribution in this context. We calculate the measurable luminosity distance in a perturbed Friedmann universe and determine
its fluctuations (within linear perturbation theory). We show that these remain smaller than one and therefore higher order perturbations are probably not relevant. The main point of our procedure is that we use only measurable quantities and not some abstract averaged expansion rate to determine the deceleration parameter. We actually calculate the luminosity distance dL(n, z) where n defines the direction of the observed supernova and z its redshift. We then determine the power spectrumCℓ(z, z′) defined by
dL(n, z) = X
ℓm
aℓm(z)Yℓm(n) (2.1)
Cℓ(z, z′) = haℓm(z)a∗ℓm(z′)i . (2.2) Here the h·i denotes a statistical average. Like for the cosmic microwave background, statistical isotropy implies that the Cℓ’s are independent ofm.
We then analyze whether the deviations of the angular diameter distance from its background value can be sufficient to fake an accelerating universe.
Aside from this problem, the new variable which is defined and calculated in this paper, might in principle present an interesting and novel observational tool to determine cosmological parameters. And this is actually the main point of our work. We hope to initiate a new observational effort, the measurement of the luminosity distance power spectrum, with this paper. A detailed numerical calculation of thedL power spectrum and the implementation of a parameter search algorithm are postponed to future work. Here we simply show that for large redshifts, z ≥ 0.4 and sufficiently high multipoles, ℓ > 10 the lensing effect dominates. However, at smaller redshift and especially at low ℓ’s other terms can become important, most notably the Doppler term due to the peculiar motion of the supernova.
The paper is organized as follows. In Section 2.2 we derive a general formula for the luminosity distance valid in (nearly) arbitrary geometries. In the next section we apply the formula to a perturbed Friedmann universe. In Section 2.4 we derive general expressions for the dL power spectrum in terms of the Bardeen potentials. We then evaluate our expressions in terms of relatively crude approximations and some numerical calculations for a simple ΩM = 1 CDM model in Section 2.5. In Section 2.6 we discuss our results and conclude.
Notation: We denote 4-vectors by arbitrary letters, sometimes with and sometimes with-out Greek indices,k= (kµ). Three-dimensional vectors are denoted bold face or with Latin indices, y = (yi). We use the metric signature (−,+,+,+). The covariant derivative of the 4-vector kin direction of the 4-vector nis often denoted by ∇nk≡(nµkα;µ).